# Pair of Opposite Sides of a Parallelogram are Equal and Parallel

Here we will discuss about one of the important geometrical property of parallelogram.

A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel

Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR.

To prove: PQRS is a parallelogram.

Construction: Join PR and QS such that they intersect at O.

Proof:

 Statement Reason In ∆OPQ and ∆ORS,1. ∠OPQ = ∠ORS 1. PQ ∥ SR and PR is a transversal. 2. ∠POQ = ∠ROS 2. Opposite angles are equal. 3. PQ = RS 3. Given. 4. ∆OPQ ≅ ∆ORSTherefore, OP = OR, OQ = OS.In ∆OPS and ∆OQR, 4. By AAS criterion of congruency.CPCTC 5. OP = OC, OQ = OS, ∠POS = ∠QOR 5. By statement 4 and reason 2. 6. ∆OPS ≅ ∆OQRTherefore, PS = QR, ∠OPS= ∠ORQ 6. By SAS criterion of congruency.CPCTC 7. PS ∥QR. 7. Alternate angles are equal. 8. PQRS is a parallelogram (Proved). 8. PQ ∥ SR and statement 7.

Corollary: In a parallelogram, each pair of opposite sides are parallel as well as equal.

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